The Language of Argument

1 4 0

C H A P T E R 6 ■ P r o p o s i t i o n a l L o g i c

Another argument form that has been historically significant is called a

hypothetical syllogism:

p ⊃ q

q ⊃ r

∴ p ⊃ r

Because we are dealing with an argument form containing three variables,

we must perform the boring task of constructing a truth table with eight

rows:

Premise Premise Conclusion

p q r p ⊃ q q ⊃ r p ⊃ r

T T T T T T OK

T T F T F F

T F T F T T

T F F F T F

F T T T T T OK

F T F T F T

F F T T T T OK

F F F T T T OK

This is fit work for a computer, not for a human being, but it is important to

see that it actually works.

Why is it important to see that these techniques work? Most people, after

all, could see that hypothetical syllogisms are valid without going through

all of this tedious business. We seem only to be piling boredom on top of

triviality. This protest deserves an answer. Suppose we ask someone why he

or she thinks that the conclusion follows from the premises in a hypothetical

syllogism. The person might answer that anyone can see that—which, by

the way, is false. Beyond this, he or she might say that it all depends on the

meanings of the words or that it is all a matter of definition. But if we go on

to ask, “which words?” and “what definitions?” then most people will fall

silent. We have discovered that the validity of some arguments depends on

the meanings of such words as “and,” “or,” “not,” and “if-then.” We have

then gone on to give explicit definitions of these terms—definitions, by the

way, that help us see how these terms function in an argument. Finally, by

getting all these simple things right, we have produced what is called a de-

cision procedure for determining the validity of every argument depending

only on conjunctions, disjunctions, negations, and propositional condition-

als. Our truth table techniques give us a mechanical procedure for settling

questions of validity in this area. In fact, truth table techniques have practi-

cal applications, for example, in computer programming. But the important

point here is that, through an understanding of how these techniques work,

we can gain a deeper insight into the notion of validity.

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